Remarks on natural deduction

MATHEMATICS

REMARKS ON NATURAL DEDUCTION BY

E. W. BETH (DEDICATED TO ROBERT FEYS)

(Communicated by Prof. A. HEYTING at the meeting of February 26, 1955)

I.

Sema'fl,tic Tableaux. - Suppose we wish to show that the formula:

(2)

(x)(y)[A(x)-+ B(y)]

is a logical consequence

1)

(1)

of the formula: (Ex)A(x) -+ (y)B(y);

then we may resort to the construction of a semantic tableau, Valid (1) (5)

(if) (7)

I

follows.

Invalid

(Ex) A (x) -+ (y) B (y) A (a) (i)

a~

(y) B (y)

(9) B(a) (10) B (b)

(2) (3) (4) (6)

(x) (y) [A (x) -+ B (y)] (y) [A (a) -+ B. (y)] A (a)-+ B (b) B(b) (i)

(if)

(Ex)A(x) (11) A (a)

(8)

Though the above tableau is fairly self-explanatory, the following remarks may be helpful. We put formula (1) as initial formula in the left column and formula (2) as initial formula in the right column. If formula (2) is to be invalid, then there must be some value of x such that (y)[A(x)-+ B(y)] is invalid; if this value is called "a", then we obtain line (3); etc. If formula (1) is to be valid, then either (i) formula (8) must be invalid, or (ij) formula (7) must be valid. The tableau clearly shows that, on account of the semantical rules for elementary logic, any attempt at constructing a suitable counter-example for proving that formula (2) is not a logical consequence of formula (1) breaks down 2).

Transformation of our Tableau into a Formal Derivation. - Let us now rearrange the formulas in our tableau in the following manner, omitting 2.

1) A. TARSKI, Ueber den Begriff der logischen Folgerung, Actes du Oongres International de PMlosophie Scientifique, fasc. VII: Logique, pp. 1-11 (Paris, 1936). 2) The tableau is constructed by a method which closely follows the semantical rules but differs from the current procedure as described, for instance, in TARSKI's Introduction to Logic, pp. 40 and 41 (2nd ed., New York, 1946).

323 the formulas (10) and (ll) which already appear under (6) and (5), respectively, and taking the formulas in the right column in the reverse order. (1)

(Ex) A (x)

(5)

A(a) (Ex) A (x) (y) B(y) B(a) B(b)

(i)

[ (8)

(if)

[ (7) (9) (6) (4) (3) (2)

~

(y) B (y) (prem)

(+ hyp 1)

A (a) ~B(b) ( - hyp 1) (y) [A (a) ~ B (y)] (x) (y) [A (x) ~ B (y)] (concl)

The resulting sequence of formulas strongly recalls a formal derivation in some System of Natural Deduction; in order to emphasise the resemblance, two horizontal lines and a few marginal notes have been added to the material taken from the original tableau, but no other changes have been made. 3. General Statement. - There is, in fact, a certain formal system F for which in a general manner the following assertion can be proved: Every semantic tableau which indicates the non-existence of a suitable counterexample for proving that a certain formula Vis not a logical consequence of certain formulas Uv U2, ••• can be rearranged such as to provide a formal derivation, in F, of the conclusion V from the premisses Uv U2, ... ; and, conversely: every formal derivation in F can be transformed into a corresponding semantic tableau. In order to prove this assertion, the following steps are required: (I) We must state precise rules for the construction of semantic tableaux; (II) We must give a precise description of the formal system F; (Ill) We must state precise rules for the transformation of a semantic tableau into a formal derivation, and vice versa; (IV) We must point out that in those cases, in which the (tentative) construction of a semantic tableau involves infinitely many steps, there is always a suitable counter-example. ad (I)

The statement of such rules presents no difficulties. Hence it will be sufficient to state those rules which are not obvious. - The appearance of (Ex)X(x) in a left column demands the introduction of a new individual p, and X(p) is then inserted in the same column; but if (Ex)X(x) appears in a right column, then X(p) is inserted in the same column for all individuals p, olQ. and new; likewise for (x)X(x). If X v Y or X-+ Y appears in a left column, or if X & Y appears in a right column, then the tableau must be split up into two subordinate tableaux. A left column and a right column which belong to the same (subordinate) tableau are called conjugated columns. If one and the same formula appears in two conjugated columns, then the (subordinate) tableau to which they belong is closed; and if both

324

subordinate tableaux of some (subordinate) tableau are closed, then that tableau itself is also closed. ad (II) This problem is now easily solved as follows: a formal derivation, in F, of a conclusion V from the premisses Uv U2 , ••• is a closed tableau in which the initial formulas are Uv U2 , ••• in the left column and V in the right column.

ad (III) In view of the above solution of (II), this problem is trivial. However, no difficulties arise if one should wish to give the derivations in F a more familiar shape. ad (IV) In this connection a difficulty arises, as an infinite tableau may present infinitely many splittings and infinitely many closures; however, the required proof results from a familiar compactness argument. 4. Final Remarks. - The above considerations are closely related to my investigations on Padoa's method 3 ) and to other work still unpublished 4 ). It will be clear that the above formal system F deserves the name of a "System of Natural Deduction" both by its similarity to the systems constructed by GENTZEN, KETONEN, ScHUTTE, KLEENE, CuRRY, QuiNE, and others, and by its close connections with a semantic approach to the problems of logic. With regard to the formal system F, the proofs of such metamathematical theorems as Herbrand's Theorem, the Theorem of Lowenheim-Skolem-Godel, Gentzen's Subformula Theorem and Extended Hauptsatz, and Bernays' Consistency Theorem are extremely simple, natural, and elegant 5 ). If, for any given formulas U1 , U2 , ••• , V, we could compute a natural number n such that, after introducing n individuals a, b, ... ,we could decide whether or not the tableau will be closed, then we would have a decision procedure for derivability. But, on account of a well-known result by A. CHURCH, there is no such decision procedure, hence in general no computation of n is possible. On the other hand, the critical number n of the individuals a, b, ... which have been introduced previously to the closure of a tableau clearly indicates the number of times we pass from a formula X(p) to a formula (x)X(x) or 3) E. W. BETH, On Padoa's Method in the Theory of Definition, these Proceedings 56 ( 1953); cf. A Topological Proof of the Theorem of Lowenheim-Skolem-Godel, ibid. 54 (1951). I take this opportUnity to repair an omission which I committed in the last mentioned paper and which was brought to my attention by the late .J. C. C. McKINSEY: a similar proof was given by ARCHIE BLAKE in his thesis: Canonical Expressions in Boolean Algebras (Chicago, 1938). 4) E. W. BETH, L'existence en mathe.matiques (forthcoming). 5) Dr. GISBERT HASENJAEGER (Miinster) has kindly informed me that related ideas have been presented in November, 1954, by K. ScHUTTE at a Symposium on Logic held in Marburg. I am also indebted to Dr. HASENJAEGER for part of what follows. I take this opportunity to express my thanks to Professor LEON HENKIN (University of California in Berkeley, now on leave in Amsterdam) for a most enlightening conversation on all the topics discussed in this communication.

325 from a formula (Ex)X(x) to a formula X(p) in the corresponding derivation, in F, of the conclusion V from the premisses Uv U2 , ••• ; hence it seems that the critical number n is an invariant which can hardly be essentially reduced if we replace the system F by a different formalisation of elementary logic (for instance, the system of Hilbert-Ackermann). Now suppose we are confronted with a somehow dubitable derivation of V from U1 , U2 , ••• in a certain formal system G. Then, on the basis of an inspection of that derivation (and, possibly, also of the formal system G), we shall be able to estimate the critical number n. And then the above method 6 ) will enable us to decide whether or not the result of the given derivation ought to be accepted. For, if our tableau is closed, then we have a correct derivation instead of the dubitable one; and if our tableau is not closed, it follows that the dubitable derivation cannot be correct; hence there can be no reason to trust its result. It will be clear that, once we have estimated the critical number n, everything else can be carried out by a machine. Another remark is concerned with Heyting's intuitionistic logic. It is well-known that, if we suitably restrict the rules of the formal system F, we obtain a corresponding intuitionistic system F*. Now suppose we have a derivation Din F, the critical number of which is n. Then the problem whether or not there is in F* an analogous derivation (that is, a derivation which starts from the same premisses and.yields the same conclusion), D*, having the same critical number n reduces to a problem concerning derivability in the intuitionistic sentential calculus and hence it can, on account of a result by Gentzen, be effectively decided. Of course, a negative result of the decision procedure does not prima facie exclude the possibility of finding in F* an analogous derivation D* having a critical number n* larger than n. But, at least practically, the situation seems to be less involved; for in concrete cases 7 ) an analysis of the semantic tableau which corresponds to the derivation D will (i) explain why it is impossible to find an analogous derivation D* having the same critical number n, and (ij) make it clear that replacing n by a larger number n* cannot make any difference. And it may even be conjectured that one can prove a general theorem to this effect; such a theorem would provide us with a decision procedure for the problem whether or not for a given derivation D in F we can find an analogous derivation D* in F*.

I nstituut voor Grondslagenonderzoek en Philosophie der Exacte W etenschappen, Universiteit van Amsterdam 6) And probably also a procedure constructed by R. STANLEY, An Extended Procedure in Quantificational Logic, Journal of Symbolic Logic 18 (1953). However, both the procedure itself and its description by the Author are so involved that I am not in a position to make a more definite statement. 7) As discussed, for instance, by S. C. KLEENE, Introduction to Metamathematics pp. 487-491 (Amsterdam-Groningen, 1952). 21 Series A